Continuous-time perpetuities and time reversal of diffusions

We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markov model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, identifies the joint law as the stationary distribution of an ergodic multi-dimensional diffusion using techniques of time reversal. This later approach allows for efficient use of Monte-Carlo simulation when estimating the distribution, as the distribution is obtained by sampling a single path of the reversed process.Comment: 42 pages; added numerical exampl

Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion

We present a two-dimensional extension of an identity in distribution due to Bougerol \cite{Bou} that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend at the level of processes, we point at further striking relations in this direction

Exponential functionals of Levy processes

This text surveys properties and applications of the exponential functional $\int_0^t\exp(-\xi_s)ds$ of real-valued L\'evy processes $\xi=(\xi_t,t\geq0)$.Comment: Published at http://dx.doi.org/10.1214/154957805100000122 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asians and cash dividends: Exploiting symmetries in pricing theory

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplace-transform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are closely related to arithmetic Asians, so that all the new techniques can be directly applied to this case.Comment: 19 pages, no figure

Efficient estimation of present-value distributions for long-dated contracts and functionals in the multivariate case

The first chapter of this thesis focuses on the problem of estimating the joint law of a discrete-time perpetuity and underlying factors which govern the cash ow rate, in an ergodic Markovian environment. Our approach is based upon the so-called time-reversal technique which allows us to identify the joint law as a stationary distribution of an ergodic multidimensional Markov chain. Furthermore, a central limit theorem (CLT) for an estimator of the joint law is provided for a specific example of the perpetuity. Our proof of the CLT rests upon the geometric ergodicity property, which is also provided and is of independent interest. We further provide a justification for the Monte Carlo methods for approximating the joint law by sampling a single path of the reversed process. The second chapter of this thesis deals with the estimation of linear functionals in multidimensional spaces. We consider two ubiquitous statistical models: a regression model with one-sided errors and a Poisson point process (PPP) model. We consider two estimation approaches: a block-wise approach, when the estimator is an aggregate of local estimators, and a maximum likelihood approach. First, we assume the regularity of the underlying function in both models to be known. We combine the block-wise approach with martingale stopping time arguments and the PPP geometry to derive the unbiased estimators. We show that the rates of convergence of the mean squared risks match the lower bounds for the risks in both models, which are also provided and are of independent interest. In the PPP model, we show that the maximum likelihood estimator is unbiased with minimal variance among all unbiased estimators. Finally, we sketch ideas for a proof of the CLT for the estimator in the PPP model in multidimensional case and provide illustrative simulations

The CTMC-Heston model: calibration and exotic option pricing with SWIFT

This work presents an efficient computational framework for pricing a general class of exotic and vanilla options under a versatile stochastic volatility model. In particular, we propose the use of a finite state continuous time Markov chain (CTMC) model, which closely approximates the classic Heston model but enables a simplified approach for consistently pricing a wide variety of financial derivatives (...